Using the results proved in [1]} for concentration of measures in product spaces, we can upper bound the \(\mathcal {H}\Delta \mathcal {H}\) -distance between \(S\)  and \(T\)  as follows:
\(\begin{aligned}d_{\mathcal {H}\Delta \mathcal {H}}(S, T)& = d_{\mathcal {H}\Delta \mathcal {H}}(S_z \times S_{\hat{z}}, T_z \times T_{\hat{z}}) \\& \le d_{\mathcal {H}\Delta \mathcal {H}}(S_z, T_z) + \int d_{\mathcal {H}\Delta \mathcal {H}}(S_{\hat{z}}|S_z, T_{\hat{z}})\mathbf {d}S_z\end{aligned}\) 
